On forward sufficient dimension reduction for categorical and ordinal responses

We introduce a forward sufficient dimension reduction method for categorical or ordinal responses by extending the outer product of gra-dients and minimum average variance estimator to categorical and ordinal-categorical generalized linear models. Previous works in this direction ex-tend forward regression to binary responses, and are applied in a pairwise manner for multi-category data, which is less efficient than our approach. Like other forward regression-based sufficient dimension reduction meth-ods, our approach avoids the relatively stringent distributional requirements necessary for inverse regression alternatives. We show the consistency of our proposed estimator and derive its convergence rate. We develop an algorithm for our methods based on repeated applications of available algorithms for forward regression. We also propose a clustering-based tuning procedure to estimate the bandwidth. The effectiveness of our estimator and related algorithms is demonstrated via simulations and applications.